Anomalous superfluid density in quantum critical superconductors

Abstract

When a second-order magnetic phase transition is tuned to zero temperature by a nonthermal parameter, quantum fluctuations are critically enhanced, often leading to the emergence of unconventional superconductivity. In these “quantum critical” superconductors it has been widely reported that the normal-state properties above the superconducting transition temperature T_c often exhibit anomalous non-Fermi liquid behaviors and enhanced electron correlations. However, the effect of these strong critical fluctuations on the superconducting condensate below T_c is less well established. Here we report measurements of the magnetic penetration depth in heavy-fermion, iron-pnictide, and organic superconductors located close to antiferromagnetic quantum critical points, showing that the superfluid density in these nodal superconductors universally exhibits, unlike the expected T-linear dependence, an anomalous 3/2 power-law temperature dependence over a wide temperature range. We propose that this noninteger power law can be explained if a strong renormalization of effective Fermi velocity due to quantum fluctuations occurs only for momenta k close to the nodes in the superconducting energy gap Δ(k). We suggest that such “nodal criticality” may have an impact on low-energy properties of quantum critical superconductors.

The physics of materials located close to a quantum critical point (QCP) are an important issue because the critical fluctuations associated with this point may produce unconventional high-temperature superconductivity (1, 2). Quantum oscillations (3, 4) and specific heat measurements (5) have shown that, in some systems, as the material is tuned toward the QCP by controlling an external parameter such as doping, pressure, or magnetic field, the effective mass strongly increases due to enhanced correlation effects. Along with this the temperature dependence of the resistivity shows a strong deviation from the standard AT² dependence in the Fermi liquid (FL) theory of metals and often shows an anomalous T-linear behavior that corresponds to the A coefficient diverging as zero temperature is approached.

Although there are many studies of non-FL behavior in the normal metallic state (1, 2), relatively little is known about how the QCP affects the superconducting properties below the critical temperature T_c. The superconducting dome often develops around the putative QCP so that when the temperature is lowered below T_c, the superconducting order parameter starts to develop and the Fermi surface becomes gapped. It is therefore natural to consider that the low-energy quantum critical fluctuations are quenched by the formation of the superconducting gap Δ, which means that the system avoids the anomalous singularities associated with the QCP. Perhaps because of this reasoning the superconducting properties are usually analyzed by the conventional theory without including temperature/field-dependent renormalization effects resulting from the proximity to the QCP. For example, in refs. 6 and 7 the NMR relaxation rate 1/T₁ in the superconducting state is fitted to the temperature dependence expected for particular gap functions with the assumption that the normal-state 1/T₁T is virtually temperature independent below T_c even when it has strong temperature dependence above T_c due to the magnetic fluctuations.

In superconductors near the QCP, the electron pairing is often unconventional with a superconducting energy gap Δ(k) that changes sign on different parts of the Fermi surface (8, 9). This sign change stems from a repulsive pairing interaction, for example resulting from antiferromagnetic spin fluctuations. In many cases, this leads to the presence of nodes in the gap Δ(k) where the gap changes sign. The low-energy excitations from the ground state in these superconductors are governed by these nodal regions. In such nodal superconductors, the effect of quantum critical fluctuations on the excited quasiparticles should be k dependent. Then the question arises as to how this effect modifies the low-energy properties in the superconducting state.

The penetration depth λ(T) is a fundamental property of the superconducting state that parameterizes the ability of a superconductor to screen an applied field by the diamagnetic response of the superconducting condensate. As fermionic quasiparticles are thermally excited from the condensate a paramagnetic current is created, which reduces the screening and increases λ. So measurements of λ(T) give direct information about density and Fermi velocity of these quasiparticles (10). When the effective mass is enhanced by the quantum critical fluctuations, the effective Fermi velocity is expected to be suppressed accordingly. In a one-component Galilean invariant superfluid, electron correlation effects may not cause the renormalization in the low-temperature penetration depth (11). In superconducting materials, however, strong electron correlations do affect the renormalization, resulting in an enhanced penetration depth, which has been reported both theoretically (12, 13) and experimentally (14, 15). To discuss the energy-dependent effect of quantum criticality on superconducting quasiparticles, the temperature dependence of penetration depth at low temperatures is thus of particular importance.

Results

Penetration Depth.

Here we begin by presenting results for the heavy-fermion system Ce_nTIn_3n+2 that is located close to a QCP (T is the transition metal element, and n is the number of CeIn₃ layers alternating with the TIn₂ blocks along the c axis). The most studied member of this series is the n = 1 member CeCoIn₅ (T_c = 2.3 K) (16), in which clear evidence for non-FL behaviors in the normal state (17, 18) and nodal superconductivity has been found (6, 19, 20). The recent discovery of superconductivity at ambient pressure in the n = 2 member Ce₂PdIn₈ (T_c = 0.68 K) (21), which exhibits very similar non-FL properties (22–27) to those in CeCoIn₅, allows a detailed comparison of the superconducting properties to extract common features in these superconductors near the antiferromagnetic QCP.

The temperature-dependent change in the in-plane penetration depth Δλ(T) = λ(T) − λ(T = 0) in both Ce₂PdIn₈ and CeCoIn₅ (Fig. 1A) exhibits strong temperature variation at low temperatures, much steeper than the flat exponential dependence expected for a fully gapped superconductor. The Δλ(T) data for Ce₂PdIn₈ are reproducible in different crystals and the data for CeCoIn₅ are fully consistent with previous studies (28–30). The strong temperature dependence indicates substantial excitations of quasiparticles at low energies, evidencing the presence of line nodes in the energy gap. This is consistent with the residual density of states (DOS) in the low-temperature limit observed by thermal conductivity (19, 23), specific heat (20, 24), and NMR measurements (6, 27). In particular, a order parameter with nodes along the 〈110〉 directions has been established in CeCoIn₅ by angle-resolved thermal conductivity (19) and specific heat measurements (20). The strong similarity between Ce₂PdIn₈ and CeCoIn₅ found in the low-temperature λ(T) points to common nodal structure in these superconductors.

Fig. 1.

Temperature dependence of the magnetic penetration depth in heavy-fermion superconductors near the antiferromagnetic QCP. (A) Low-temperature changes in the magnetic penetration depth Δλ(T) = λ(T) − λ(T = 0) of single crystals of Ce₂PdIn₈ and CeCoIn₅. The curves are shifted vertically for clarity and the data for CeCoIn₅ are multiplied by 3.5. Inset shows the ac susceptibility over the whole temperature range measured by the frequency shift of the tunnel diode oscillator, showing sharp superconducting transitions. The dashed line is an exponential temperature dependence expected for a fully gapped s-wave superconductor. (B) The same data plotted against (T/T_c)². The solid line represents a T² dependence. (C) The same data plotted against (T/T_c)^3/2. The solid line represents a T^3/2 dependence.

In a pure d-wave superconductor with line nodes such as high-T_c cuprates, it is well established that Δλ(T) shows a linear T dependence at low temperatures, which stems from the linear energy dependence of low-energy DOS of quasiparticles. However, a clear deviation from the T-linear dependence is observed in the present heavy-fermion superconductors (Fig. 1A). The data also strongly deviate from the T² dependence expected for the dirty limit case (31) (Fig. 1B). We rather find that the power-law dependence T^α with an unusual exponent α = 3/2 can describe the observed low-temperature variation in a wide range in both superconductors (Fig. 1C).

A few explanations for the superlinear temperature dependence of Δλ(T) in d-wave superconductors have been put forth, including the effect of impurity scattering (31), nonlocal effect near the nodes (32), and phase fluctuations (33). Often an interpolation formula describing a crossover from T to T² dependence Δλ(T) ∝ T²/(T + T*) is used to describe the experimental data. In particular, the impurity effect leads to the crossover temperature , where Γ is the impurity scattering rate and Δ₀ is the maximum gap. This successfully accounts for the systematic change of Δλ(T) with impurity scattering observed in Zn-doped YBa₂Cu₃O_6.95 (34). In the present heavy-fermion case, however, fitting to this crossover formula in these relatively clean superconductors yields T*/T_c values (∼0.5 for Ce₂PdIn₈ and ∼0.2 for CeCoIn₅) substantially larger than the estimates from these theories (SI Text, Comparisons with Existing Theories and ref. 30). Therefore, the T^3/2 dependence in a wide T/T_c range commonly observed in these superconductors with quite different T_c, which is distinctly different from the T-linear dependence in, e.g., YBa₂Cu₃O_6.95 (34), rather suggests some inherent mechanism related to their closeness to the antiferromagnetic QCP.

Superfluid Density.

To discuss the precise temperature evolution of quasiparticle excitations, we analyze our data in terms of the superfluid density 1/λ²(T). We use the reported values of λ(0) [280 nm for CeCoIn₅ (30) and 1,010 nm for Ce₂PdIn₈ (35)], from which the normalized superfluid density ρ_s(T) = λ²(0)/λ²(T) has been obtained (Fig. 2A). The factor of ∼3.5 difference in the slope of dλ/d(T/T_c) (Fig. 1A) is consistent with the difference in λ(0) in these two superconductors, which is also consistent with the larger γ-value in Ce₂PdIn₈ (24). This results in an almost collapse of the full temperature dependence ρ_s(T) into a single curve (Fig. 2A, Inset), and the low-temperature variation shows the (T/T_c)^3/2 dependence with nearly identical slopes.

Fig. 2.

Universal T^3/2 dependence of superfluid density in unconventional superconductors in the vicinity of the antiferromagnetic order. (A) The normalized superfluid density ρ_s as a function of (T/T_c)^3/2 at low temperatures for Ce₂PdIn₈ and CeCoIn₅. The lines represent T^3/2 dependence. Inset shows the overall temperature dependence up to (T/T_c) = 1. (B) A similar plot for iron-pnictide superconductor BaFe₂(As_0.7P_0.3)₂ and organic superconductor κ-(ET)₂Cu[N(CN)₂]Br. The solid lines are the fits to T^3/2 dependence. The low-temperature data for BaFe₂(As_0.7P_0.3)₂ are vertically shifted for clarity.

This (T/T_c)^3/2 dependence can be also seen in other classes of materials that are close to antiferromagnetic order. In Fig. 2B we show data for the organic superconductor κ-(BEDT-TTF)₂Cu[N(CN)₂]Br, where BEDT-TTF (abbreviated as ET hereafter) denotes bisethylenedithio-tetrathiafulvalene, which are consistent with but measured to lower temperature than that in a previous report (36) and in the iron-pnictide superconductor BaFe₂(As_0.7P_0.3)₂ (15). Both these materials are known to have line nodes in their superconducting gap (37, 38). In the BaFe₂(As_1−xP_x)₂ series, there is clear evidence for the antiferromagnetic QCP being located at x ∼ 0.30 (15). In κ-(ET)₂Cu[N(CN)₂]Br, although the proposed phase diagram suggests that the boundary between the superconducting and antiferromagnetic states is a first-order phase transition (39), the anomalous critical exponent near the Mott critical end point (40) suggests the presence of strong antiferromagnetic quantum fluctuations. The normalized superfluid density ρ_s(T) in these superconductors shows very similar (T/T_c)^3/2 dependence at low temperatures with slight deviations at the lowest temperatures below T/T_c ∼ 0.05 ((T/T_c)^3/2 ≲ 0.01).

These results imply that the (T/T_c)^3/2 dependence of superfluid density in a wide T/T_c range is a robust property in unconventional superconductors, in which strong antiferromagnetic fluctuations are present (for comparisons between iron-pnictides and cuprates, see SI Text, Comparisons Between Iron-Pnictides and Cuprates).

Discussion

If the quantum fluctuations survive the Fermi-surface gapping, the effective mass is expected to diverge in the zero-temperature limit. In such a case, the strong enhancement of mass leads to a reduction of superfluid density when approaching the zero-temperature limit. It has been recently found in iron-pnictides that the zero-temperature superfluid density 1/λ²(0) shows a strong reduction at the QCP (15), indicating the strong quantum critical fluctuations directly affect the superconducting condensate. However, the temperature dependence of the normalized superfluid density ρ_s(T) actually continues to rise with decreasing temperature (Fig. 2 A and B, Insets). This suggests that the effect of temperature-dependent quantum fluctuations enters as corrections in the temperature dependence of superfluid density at low temperatures. The anomalous noninteger power-law temperature dependence of ρ_s(T) universally observed in quantum critical superconductors thus calls for further theoretical understanding.

Nodal Quantum Criticality.

We propose a possible scenario that this universal behavior can be naturally explained by invoking a strong momentum dependence of renormalization due to the nodal gap structure. Suppose that the gap formation below T_c quenches the low-energy quantum critical fluctuations, which prevents the effective mass enhancement in the superconducting state. Then in the nodal superconductors, because of the strong momentum dependence of the gap magnitude, we may consider that the degree of quenching of the quantum fluctuations in the superconducting state has a strong momentum dependence as well; the low-energy fluctuations are expected to be strongest near the nodes where the Fermi surface is not gapped (Fig. 3A). To model this effect, we consider the angle dependence of the effective Fermi velocity along the underlying Fermi surface. We consider a simple cylindrical Fermi surface and assume that the renormalization in is inversely related to the enhancement in the effective mass m*(k) (Fig. 3B). [Here is given by the dynamic effective mass that is different from, but closely related to, the thermodynamic mass (12).] The effective mass enhancement on approaching the QCP can be described by m*² ∝ (p − p_QCP)^−β, where p is a nonthermal parameter controlling the distance from the QCP at p_QCP. The critical exponent β has been estimated experimentally by using magnetic fields as the parameter p, and in several materials a value of β close to unity has been reported (41, 42). In the present case, we take the magnitude of the superconducting gap |Δ| as the control parameter, because the gap magnitude determines the degree of quenching of low-energy fluctuations. We thus assume , which allows us to calculate the temperature dependence of superfluid density by the integral over the Fermi surface S as (10, 13)

Fig. 3.

Nodal quantum criticality in unconventional superconductors. (A) The momentum-dependent gap Δ(k) (whose magnitude is illustrated by thin lines with gray shading) opens on the Fermi surface (thick line) and has nodes (red circles) at certain directions. In -wave superconductors, for example, Δ(k) has strong in-plane anisotropy Δ₀ cos(2ϕ) as a function of azimuthal angle ϕ. In quasi-2D systems, the Fermi surface is approximated by a cylinder, and thus the gap has nodal lines perpendicular to the planes. At the nodes, the gap is zero and thus the quantum critical fluctuations may be present (red shading) on the ungapped Fermi surface. (B) The nodal quantum fluctuations lead to the momentum dependence of the renormalization in near the nodes (blue lines). (C) The angle dependence of the renormalized Fermi velocity relative to the unrenormalized one, v_F along the Fermi surface, assumed for calculations of the superfluid density in D. Near the nodes, we illustrate different cutoff levels, which model finite distances from the QCP or disorder. (D) Calculated normalized superfluid density as a function of (T/T_c)^3/2 with different cutoff levels, which explains the deviation from the T^3/2 dependence at very low temperatures. Inset is the full temperature dependence up to (T/T_c) = 1.

where (v_F) is the effective velocity in the superconducting (normal) state, the subscripts i, j denote the directions of the current and vector potential (we take both along a), is Yosida function, and f(E) is the Fermi–Dirac function for the quasiparticle energy E. By using the formula of Δ(k) = Δ₀ cos(2ϕ) and β = 1, the normalized superfluid density ρ_s is calculated with (Fig. 3C), which results in the T^3/2 dependence at low temperatures (Fig. 3D).

As real materials will never be located exactly at the QCP, there will be a cutoff for the diverging m* near the nodes. This can be modeled by introducing a minimum value for . This leads to an upward deviation from the T^3/2 dependence of ρ_s at very low temperatures approaching T-linear behavior at sufficiently low temperatures (Fig. 3D). This can explain the essential features of the experimental observations. We note that the presence of the cutoff can be also expected even at the exact QCP, because the dynamical susceptibility in the zero-temperature limit should diverge only on certain regions of the Fermi surface (determined by the momentum dependence of the spin fluctuations), which in general may be different from the nodal points (9). Of course, disorder would also produce some additional changes to the temperature dependence but we have not included this in the present model.

An additional factor could also come from the temperature dependence of the renormalization of m*. As only the thermally excited quasiparticles will be renormalized, the angular range near the nodes where this occurs [i.e., where Δ(k) < k_BT] is quite limited at low temperature, so this has a rather minor effect. In our temperature range of interest, T ≲ 0.2T_c, this region is actually limited to a narrow angle range of ∼ ±3° near the nodes. If we add a cutoff constant of 0.3 covering this angle range (Fig. 3C), we find that the T dependence is affected only in the lowest-temperature range of (T/T_c)^1.5 ρ 0.01, above which the T^3/2 dependence of ρ_s still holds (Fig. 3D). This exercise implies that the inclusion of the temperature dependence of renormalization will not change ρ_s(T) significantly in the temperature range of interest (for more discussion, see SI Text, Temperature Dependence of the Renormalization).

We also note that in iron-pnictides the gap symmetry is most likely s-wave (43), and the model based on the d-wave gap may not be applicable. However, the fundamental physics that the low-energy excitations are governed by the nodal regions should be essentially the same. Although the detailed structure of the momentum-dependent Fermi velocity (such as the precise value of the critical exponent β) will affect the detailed ρ_s(T) (SI Text, Effect of Quantum Critical Exponent), it is striking that such a simple model can capture the salient feature of the unusual behavior of quasiparticle excitation in the superconducting state of these materials. In an FL theory, the renormalization of the effective Fermi velocity in Eq. 1 can be described by the interplay between k-dependent quasiparticle interaction and nodal gap structure (13). More detailed theoretical calculations will be needed to account for strong energy dependence of critical fluctuations, the effects of disorder [such as quasiparticle scattering interference (44)], and the inelastic quasiparticle scattering rate.

We suggest that the nodal quantum criticality proposed here is an important aspect of unconventional superconductivity close to the magnetic QCP. In addition to the penetration depth analyzed here there will be implications for most other superconducting properties such as thermal conductivity and the NMR relaxation rate 1/T₁(T) that has been long known to have strong deviations from the T³ law, which in the usual analysis gives overestimates of the residual quasiparticle DOS in quantum critical superconductors (6, 7). It should be straightforward to extend our analysis to these other properties.

Materials and Methods

The magnetic penetration depth measurements have been made in Kyoto for high-quality single crystals of Ce₂PdIn₈ and CeCoIn₅ (grown by the self-flux method) down to ∼60 mK by using a tunnel diode oscillator with the resonant frequency of ∼13 MHz (SI Text, Sample Information and Experimental Techniques). The weak ac field is applied along the c axis, which excites supercurrents in the ab plane. The measurements in κ-(BEDT-TTF)₂Cu[N(CN)₂]Br have been done in Bristol by the same technique. Here the supercurrents are in the conducting planes.